Finally, let's consider the case of an ensemble of quantum systems, starting in a pure state represented by the density matrix \(\rho _0\) . It's usual to write the evolution of the density matrix using its commutator with the Hamiltonian; here, however, it is more convenient to work in the so-called Fock-Liouville space, in which the density matrix can be written as a vector, \(\rho _0 \rightarrow \left|\left.\rho _0 \right>\right>\) , and the Liouvillian superoperator as a matrix \(\tilde{\mathcal {L}}\)  [1]}. Then the evolution in presence of a randomly varying Hamiltonian (and thus, Liouvillian), can be written as
\(\left|\left.\rho \right>\right> (t) = \int d\psi (t)p[\psi (t)]e^{\int _0^t\tilde{\mathcal {L}}(\tau )d\tau } \left|\left.\rho _0 \right>\right>\) 
